On the Boltzmann equation for Fermi–Dirac particles with very soft potentials: Global existence of weak solutions

Abstract

For general initial data we prove the global existence and weak stability of weak solutions of the Boltzmann equation for Fermi–Dirac particles in a periodic box for very soft potentials ( − 5 < γ ⩽ − 3 ) with a weak angular cutoff. In particular the Coulomb interaction ( γ = − 3 ) with the weak angular cutoff is included. The conservation of energy and moment estimates are also proven under a further angular cutoff. The proof is based on the entropy inequality, velocity averaging compactness of weak solutions, and various continuity properties of general Boltzmann collision integral operators.